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The Three Magical Boxes

Q: You are playing a game wherein you are presented 3 magical boxes. Each box has a set probability of delivering a gold coin when you open it. On a single attempt, you can take the gold coin and close the box. In the next attempt you are free to either open the same box again or pick another box. You have a 100 attempts to open the boxes. You do not know what the win probability is for each of the boxes. What would be a strategy to maximize your returns?

A: Problems of this type fall into a category of algorithms called "multi armed bandits". The name has its origin in casino slot machines wherein a bandit is trying to maximize his returns by pulling different arms of a slot machine by using several "arms". The dilemma he faces is similar to the game described above. Notice, the problem is a bit different from a typical estimation exercise. You could simply split your 100 attempts into 3 blocks of 33,33 & 34 for each of the boxes. But this would not be optimal. Assume that one of the boxes had just a \(1\%\) probability of yielding a golden coin. Even as you probe and explore that box you know intuitively that you have spent a fair amount of attempts to simply reinforce something you already knew. You need a strategy that adjusts according to new information that you gain from each attempt. Something that gradually transitions away from a box that yields less to a box that yields more.

Assume at the beginning of the game you do not know anything about the yield probabilities. Assign a prior set of values of \(\big[\frac{1}{2}, \frac{1}{2},\frac{1}{2}\big]\). Simultaneously maintain a set of likelihoods using which you will decide which box to sample next. Initially all three values are set to 1s \(\{p_1 = 1,p_2 = 1,p_3 = 1\}\). First open the boxes in succession and use up \(n\) attempts per box. If you denote the number of successes for each box as \(\{s_1,s_2,s_3\}\), then you could update the posterior distribution of your belief in what box yields as follows
$$
p_1 = \frac{1 + s_1}{2 + n} \\
p_2 = \frac{1 + s_2}{2 + n} \\
p_3 = \frac{1 + s_3}{2 + n}
$$
Think of this as your initializing phase. Once you initialize your estimates, subsequent choice of boxes should be based on a re-normalized probability vector derived from \(p_1,p_2,p_3\). What this means is that the probability you would pick a box is computed as follows
$$
P(\text{pick box 1}) = \frac{p_1}{p_1 + p_2 + p_3} \\
P(\text{pick box 2}) = \frac{p_2}{p_1 + p_2 + p_3} \\
P(\text{pick box 3}) = \frac{p_3}{p_1 + p_2 + p_3}
$$
What ends up happening here is that you will pick the box which has the highest probability of winning based on information gleaned up to a certain point. Another benefit of this approach is you are learning in real time. If a certain box isn't yielding as much as another you don't discard opening that box all together, instead you progressively sample it less often.

If you are looking to buy some books in probability here are some of the best books to own

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.